Properties

Label 6-7800e3-1.1-c1e3-0-10
Degree $6$
Conductor $474552000000$
Sign $-1$
Analytic cond. $241610.$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 7-s + 6·9-s − 3·11-s + 3·13-s + 3·17-s − 6·19-s + 3·21-s − 10·27-s − 29-s − 7·31-s + 9·33-s + 14·37-s − 9·39-s − 4·43-s − 47-s − 13·49-s − 9·51-s + 5·53-s + 18·57-s − 7·59-s − 5·61-s − 6·63-s + 3·67-s − 10·71-s − 10·73-s + 3·77-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s − 0.904·11-s + 0.832·13-s + 0.727·17-s − 1.37·19-s + 0.654·21-s − 1.92·27-s − 0.185·29-s − 1.25·31-s + 1.56·33-s + 2.30·37-s − 1.44·39-s − 0.609·43-s − 0.145·47-s − 1.85·49-s − 1.26·51-s + 0.686·53-s + 2.38·57-s − 0.911·59-s − 0.640·61-s − 0.755·63-s + 0.366·67-s − 1.18·71-s − 1.17·73-s + 0.341·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(241610.\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
13$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + T + 2 p T^{2} + 17 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T + 16 T^{2} + 13 T^{3} + 16 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 + 6 T + 37 T^{2} + 128 T^{3} + 37 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 37 T^{2} - 44 T^{3} + 37 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + T + 34 T^{2} - 35 T^{3} + 34 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 7 T + 28 T^{2} + 9 T^{3} + 28 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 14 T + 167 T^{2} - 1096 T^{3} + 167 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 39 T^{2} - 52 T^{3} + 39 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 4 T + 125 T^{2} + 332 T^{3} + 125 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + T + 58 T^{2} + 5 T^{3} + 58 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 5 T + 34 T^{2} + 251 T^{3} + 34 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 7 T + 142 T^{2} + 835 T^{3} + 142 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 5 T + 138 T^{2} + 601 T^{3} + 138 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 3 T + 184 T^{2} - 417 T^{3} + 184 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 10 T + 201 T^{2} + 1240 T^{3} + 201 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 159 T^{2} + 1512 T^{3} + 159 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 16 T + 313 T^{2} + 2628 T^{3} + 313 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 11 T + 238 T^{2} - 1571 T^{3} + 238 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 8 T + 235 T^{2} + 1376 T^{3} + 235 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 26 T + 487 T^{2} - 5484 T^{3} + 487 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.27170784948683271495538723535, −6.98372272803536800573515052568, −6.67014190930989474570741717556, −6.59422264044992996861628876077, −6.22049779497179497004066467861, −6.17200063863426957627864504018, −5.90708745996603064955019558790, −5.59037195374214280462846905440, −5.50779935593322518707527460864, −5.39702807244197630443032859618, −4.85806711340994013905219895019, −4.71607043135361251925842993270, −4.49476337979363732878875839719, −4.30220810689635695705442641791, −3.98963663106736094084059395411, −3.78365514397281470597522610012, −3.30421289677450391591983877407, −3.18289113047586463420708106926, −2.96639318747627023695066468304, −2.35329654145435440488806614057, −2.29551530298807110433205226926, −1.88696621597137468217732814057, −1.40870956127180496372548270338, −1.14494092282001473116443733280, −1.02624521969201650089672914279, 0, 0, 0, 1.02624521969201650089672914279, 1.14494092282001473116443733280, 1.40870956127180496372548270338, 1.88696621597137468217732814057, 2.29551530298807110433205226926, 2.35329654145435440488806614057, 2.96639318747627023695066468304, 3.18289113047586463420708106926, 3.30421289677450391591983877407, 3.78365514397281470597522610012, 3.98963663106736094084059395411, 4.30220810689635695705442641791, 4.49476337979363732878875839719, 4.71607043135361251925842993270, 4.85806711340994013905219895019, 5.39702807244197630443032859618, 5.50779935593322518707527460864, 5.59037195374214280462846905440, 5.90708745996603064955019558790, 6.17200063863426957627864504018, 6.22049779497179497004066467861, 6.59422264044992996861628876077, 6.67014190930989474570741717556, 6.98372272803536800573515052568, 7.27170784948683271495538723535

Graph of the $Z$-function along the critical line